Can triangles help spot a bomb?

Airport security staff have a daunting task. With impatient queues
looming over them they need to search x-ray scans of cluttered
suitcases for several items at once: knives, guns and bombs. How can we ease their task and make sure they don't miss a crucial item? To find out, scientists at the University of Southampton are trying to understand how we humans take in visual information. They are presenting their work at this year's Royal Society Summer Science Exhibition 2011, which runs from the 5th to the 10th of July. The humble triangle plays a crucial role in the experiments they perform.

Diagram 1

Eye tracking experiments record the eye movements of volunteers
looking at an image on a screen to see what kind of visual features their eyes latch onto. It's important to know how big the visual stimuli on the screen are, but size is in the eye
of the beholder. Even the biggest things look small when you're far
enough away.

A clever way of taking account of this is to measure size
in terms of the visual
angle. The length of an object in an image, for example a knife, is measured in terms
of the angle formed by the two lines starting at your eye and ending at either end of the object (see diagram 1). Using the visual angle to measure size, scientists can accurately compare results of different experiments that may have been run in laboratories with different sized stimuli and different viewing distances.

But how can you work out the visual angle from the actual
size of an object on a screen and vice versa?

Diagram 2

Suppose the viewer’s gaze is directed right at the mid-point of the object on the screen, as shown in diagram 2. We write for the visual angle, for the length of the object on the screen and for the viewing distance. The triangle has a right angle at the vertex . The angle at vertex is .

A little trigonometry tells us that

Given an image of an object of length , we can now work out how far we need to place the viewer from the screen to make sure they see the object at a visual angle :

So if and , the necessary viewing distance is

Similarly, given a fixed viewing distance we can work out how big we need to make the image of the object to get a visual angle :

For and we get

An eye tracking experiment.

Finally, given a fixed viewing distance and an object of length , we can work out the visual angle as

So an image with viewed at a distance of gives a visual angle of

The visual angle is the standard way of measuring size in vision science, whether it's about improving airport security or understanding the psychology or biology of vision. It's also a favourite with astronomers, who for centuries have been using visual angles to describe the size of the Moon, the Sun and other celestial objects as viewed from Earth.

Comments

I didn't really get the connection between the airport security staff and the visual angle mathematics? How does knowledge of the equations of the visual angle assist security staff in any way? After the opening paragraph there was no further mention of airport security or how this maths was of use in practical applications.

I normally enjoy Plus articles very much, but unless I missing something obvious (in which case apologies!), this one doesn't quite follow through on its premise.

Sorry if it's not clear - the visual angle is used in all sorts of research into vision. The researchers of this particular exhibit use it to find out how we look at items on a computer screen. Their results help them devise better ways of displaying the information from x-rays, for example through 3D screens, which will make airport staff's task of searching for several kinds of items at the same time easier.